The universal order one invariant of framed knots in mostS1–bundles over orientable surfaces

Vladimir V Chernov
2003 Algebraic and Geometric Topology  
It is well-known that self-linking is the only Z valued Vassiliev invariant of framed knots in S^3. However for most 3-manifolds, in particular for the total spaces of S^1-bundles over an orientable surface F not S^2, the space of Z-valued order one invariants is infinite dimensional. We give an explicit formula for the order one invariant I of framed knots in orientable total spaces of S^1-bundles over an orientable not necessarily compact surface F not S^2. We show that if F is not S^2 or S^1
more » ... F is not S^2 or S^1 X S^1, then I is the universal order one invariant, i.e. it distinguishes every two framed knots that can be distinguished by order one invariants with values in an Abelian group.
doi:10.2140/agt.2003.3.89 fatcat:jl2vs2um7fernjlfkuzgr4ktaq